1

Chapter IV: Electronic Spectroscopy of diatomic molecules

IV.2.1 Molecular orbitals

IV.2.1.1. Homonuclear diatomic molecules The molecular orbital (MO) approach to the electronic structure of diatomic

molecules is not unique but it lends itself to a fairly qualitative description, which

we require here. It consists in considering the two nuclei, without their electrons, a distance

apart equal to the equilibrium internuclear distance and constructing MOs

around them – just as we construct atomic orbitals (AOs) around a single bare

nucleus. Electrons are then fed into the MOs in pairs (with the electron spin

quantum number ms=±1/2), just as for atoms, along the Aufbau principle, to

give the ground configuration of the molecule.

The basis of constructing the MOs is the linear combination of atomic orbitals

(LCAO) method. This accounts for the fact that, in the region close to a nucleus,

the MO wave function resembles an AO wave function for the atom of which the

nucleus is a part. It is reasonable, then, to express a MO wave function ψ as a

linear combination of AO wave functions χi on both nuclei:

∑ (IV.1)

where ci is the coefficient of the wave function χi. However, not all linear

combinations are effective in the sense of producing an MO which is appreciably

different from the AOs from which it is formed. For effective linear

combinations:

Condition 1: The energies of the AOs must be comparable.

Condition 2: The AOs should overlap as much as possible.

Condition 3: The AOs must have the same symmetry properties with respect to

certain symmetry elements of the molecule.

For a homonuclear diatomic molecule with nuclei 1 and 2, the LCAO gives the

MO wave function

(IV.2)

For example for the N2 molecule, the nitrogen 1s AOs satisfy Condition 1, since

their energies are identical, but not Condition 2 because the 1s AOs are close to

the nuclei, resulting in little overlap. In contrast, the 2s AOs satisfy both

conditions and since they are spherically symmetrical, Condition 3 as well.

Examples of AOs which satisfy Conditions 1 and 2 but not Condition 3 are the 2s

and 2px orbitals shown in Figure IV.1. The 2s AO is symmetric to reflection across

a plane containing internuclear z-axis and perpendicular to figure, but 2px AO is

not.

Important properties of the MO are the energy E and value of c1 and c2 in

Equation IV.2. They are obtained from the Schrödinger equation

(IV.3)

Multiplying both sides by ψ *, the complex conjugate of ψ and integrating over all

space gives ∗

∗ (IV.4)

E can be calculated only if ψ is known, so what is done is to make an intelligent

guess at the MO wave function, say ψn, and calculate the corresponding value of

the energy En from Equation (IV.4). A second guess at ψ, say ψm , gives a

corresponding energy E m. The variation principle states that if nm EE then

ψm is closer than ψn to the true MO gs wave function. In this way the true

ground state wave function can be approached as closely as we choose but this

is usually done by varying parameters in the chosen wave function until they

have their optimum values.

Combining Equations (IV.2) and (IV.4) and assuming that 1 and χ2 are not

complex gives

3

(IV.5)

If the AO wave functions are normalized

1 (IV.6)

and, as H is a hermitian operator,

(IV.7)

The quantity

(IV.8)

is called the overlap integral as it is a measure of the degree to which χ1 and χ2 overlap each other. In addition, integrals such as dtH 21 are abbreviated to H11. All this reduces

Equation (IV.5) to:

(IV.9)

Using the variation principle to optimize c1 and c2 we obtain 1c

E

and 2c

E

from

Equation (IV.9) and put them equal to zero, giving:

00

(IV.10)

Where we have replaced E by E since, although it is probably not the true

energy, it is the nearest approach to it with the wave function of Equation (IV.2).

Equations (IV.10) are the secular equations and the two values of E which satisfy

them are obtained by solution of the secular determinant:

0 (IV.11)

H12 is the resonance integral, usually symbolized by . In a homonuclear diatomic

molecule H11 = H22 = , which is known as the Coulomb integral, and the

secular determinant becomes:

0 (IV.12)

giving

0 (IV.13)

from which we obtain two values of E,

(IV.14)

If we are interested only in very approximate MO wave functions and energies

we can assume that S = 0 (a typical value is about 0.2) and that the hamiltonian H

is the same as in the atom giving the atom, giving =EA, the AO energy. These

assumptions result in

≅ (IV.15

At this level of approximation the two MOs are symmetrically displaced from

EA with a separation of 2. Since is a negative quantity the orbital with energy

(EA+) lies lowest, as shown in Figure IV.2.

With the approximation S=0, the secular equations become

00

(IV.16)

Putting E= E+ or E- we get c1/c2= 1 or -1, resp., and therefore the

corresponding wave functions ψ+ and ψ- are given by:

5

)(

)(

21

21

N

N (IV.17)

N+ and N- are normalization constants obtained from the conditions:

122 dd (IV.18)

Neglecting d21, the overlap integral, gives N+=N-=2-1/2 and

ψ 2 (IV.19)

In this way every linear combination of two identical AOs gives two MOs, one

higher and the other lower in energy than the AOs. Figure IV.3 illustrates the

MOs from 1s, 2s and 2p AOs showing the approximate forms of the MO wave

functions.

The designation g1s, u1s, ..., includes the AO from which the MO was

derived, 1s in these examples, and also a symmetry species label, here g or u

for the MO from the D∞h point group. However, use of symmetry species labels

may be more or less avoided by using the fact that only -type and -type MOs

are usually encountered, and these can easily be distinguished by the orbitals

being cylindrically symmetrical about the internuclear axis whereas the orbitals

are not: this can be confirmed from the examples of orbitals in Figure IV.3.

The ‘g’ or ‘u’ subscripts imply symmetry or antisymmetry, respectively, with

respect to inversion through the centre of the molecule but are often dropped in

favour of the asterisk as in, say, *2s or *2p. The asterisk implies antibonding

character due to the nodal plane, perpendicular to the internuclear axis, of such

orbitals: those without an asterisk are bonding orbitals.

The MOs from 1s, 2s and 2p AOs are arranged in order of increasing energy in

Figure IV.4, which is applicable to all first-row diatomic molecules except O2 and

F2. Because of the symmetrical arrangement, illustrated in Figure IV.2, of the two

MOs with respect to the AOs, and because the resonance integral for the MO

formed from the 2pz AOs is larger than for those formed from 2px and 2py MOs,

the expected order of MO energies is:

1 ∗1 2 ∗2 2 2 ∗2 ∗2

(IV.20)

In fact, this order is maintained only for O2 and F2. For all the other first-row

diatomic molecules g 2s and g 2p interact (because they are of the same

symmetry) and push each other apart such that g 2p is now above u 2p giving

the order

1 ∗1 2 ∗2 2 2 ∗2 ∗2

(IV.21)

This is the order shown in Figure IV.4.

The electronic structure of any first-row diatomic molecule can be obtained by

feeding the available electrons into the MOs in order of increasing energy and

taking account of the fact that orbitals are doubly degenerate and can

accommodate four electrons each. For example, the ground configuration of the

14-electron nitrogen molecule is

1 ∗1 2 ∗2 2 2

(IV.22)

There is a general rule that the bonding character of an electron in a bonding

orbital is approximately cancelled by the antibonding character of an electron

in an antibonding orbital. In nitrogen, therefore, the bonding of the two

electrons in g 1s is cancelled by the antibonding of two electrons in *u 1s and

similarly with g 2s and *u 2s. So there remain six electrons in the bonding

orbitals u 2p and g 2p and, since

bond order = half the net number of bonding electrons,

the bond order is three, consistent with the triple bond of N2.

The ground configuration of O2

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1 ∗1 2 ∗2 2 2 ∗2 (IV.23)

is consistent with a double bond. Just as for the ground configuration of atoms the

first of Hund’s rules applies and, if several states arise from the gs

configuration, the l o w e s t ( ground) state has the highest multiplicity. In O2,

the two electrons in the unfilled g 2p orbital may have their spins parallel,

giving S =1 ( multiplicity of 3), or S =0 (multiplicity of 1).

Therefore, the ground state of O2 is a triplet state and the molecule is

paramagnetic. Singlet states with the same configuration as in Equation (IV.23)

with antiparallel spins of the electrons in the *g 2p orbital are higher in energy

and from the low-lying excited states.

Fluorine has the ground configuration

1 ∗1 2 ∗2 2 2 ∗2

(IV.24)

consistent with a single bond.

Just as for atoms, excited configurations of molecules are likely to give rise to

more than one state. For example the excited configuration

1 ∗1 2 ∗2 2 2

(IV.25)

of the short-lived molecule C2 results in a triplet and a singlet state because the two

electrons in partially filled orbitals may have parallel or antiparallel spins.

IV.2.1.2 Heteronuclear diatomic molecules

Some heteronuclear diatomic molecules, such as nitric oxide (NO), carbon

monoxide (CO) and the short-lived CN molecule, contain atoms which are

sufficiently similar that the MOs resemble quite closely those of homonuclear

diatomics. In NO, the 15 electrons can be fed into MOs, in the order relevant to

O2 and F2, to give the ground configuration

1 ∗1 2 ∗2 2 2 ∗2

(IV.26)

(for heteronuclear diatomics, the ‘g’ and ‘u’ subscripts given in Figure IV.4 must

be dropped as there is no centre of inversion, but the asterisk still retains its

significance.) The net number of bonding electrons is five giving a bond order of

2. This configuration gives rise to a doublet state, because of the unpaired

electron in the *2p orbital, and paramagnetic properties.

Even molecules such as the short-lived SO and PO molecules can be treated, at the

present level of approximation, rather like homonuclear diatomics. The reason

being that the outer shell MOs can be constructed from 2s and 2p AOs on the

oxygen atom and 3s and 3p AOs on the sulphur or phosphorus atom and appear

similar to those shown in Figure IV.3. These linear combinations, such as 2p on

oxygen and 3p on sulphur, obey all the conditions given on page 1 – notably

Condition 1, which states that the energies of the AOs should be comparable.

There is, in principle, no reason why linear combinations should not be made

between AOs which have the correct symmetry but very different energies, such

as the 1s orbital on the oxygen atom and the 1s orbital on the phosphorus atom.

The result would be that the resonance integral (Figure IV.2) would be

extremely small so that the MOs would be virtually unchanged from the AOs and

the linear combination would be ineffective.

In a molecule such as hydrogen chloride (HCl) the MOs bear no resemblance

to those shown in Figure IV.3 but the rules for making effective linear

combinations still hold good. The electron configuration of the chlorine atom is

KL3s23p5 and it is only the 3p electrons (ionization energy 12.967 eV), which

are comparable in energy with the hydrogen 1s electron (ionization energy

13.598 eV). Of the 3p orbitals it is only 3pz which has the correct symmetry

for a linear combination to be formed with the hydrogen 1s orbital, as Figure

9

IV.5(a) shows. The MO wave function is of the form in Equation (IV.2) but c1 =c2

is no longer ±1. Because of the higher electronegativity of Cl compared with

H there is considerable electron concentration near Cl, which go into the resulting

bonding orbital. The two electrons in this orbital form the single bond.

The 3px and 3py AOs of Cl cannot overlap with the 1s AO of H and the electrons

in them remain as lone pairs in orbitals which are very little changed in the

molecule, as shown in Figure IV.5(b). In fact, we can easily see that the overlap

between 2s and 2px cancels out. We start appreciating how we can use symmetry

arguments to say something about our molecule without actually solving the

problem. Consider, for example a 2s orbital on the C atom and a 2px orbital on

the O atom (Figure IV.1). If we turn the molecule around by 180 degrees about its

axis, the molecule looks the same to us, the Hamiltonian has remained the same

and the 2s orbital has remained the same too. The 2px orbital, on the other hand

has changed sign. So the integrals H12 = <2s|H|2px> and S = <2s|2px> also must

change sign.

IV.2.2 Classification of electronic states

Here we only consider the non-rotating molecule, so we are concerned only

with orbital and spin angular momenta of the electrons. As in polyelectronic

atoms the orbital and spin motions of each electron create magnetic moments

which behave like small bar magnets. The way in which the magnets interact

with each other represents the coupling between the electron motions.

For all diatomic molecules, the best approximation for the SO coupling

is the Russell–Saunders approximation (§ IV.1.2.3). The orbital angular

momenta of all electrons are coupled to give L

and all the electron spin

momenta to give S

. However, if there is no highly charged nucleus in the

molecule, the spin–orbit coupling between L

and S

is sufficiently weak that,

instead of being coupled to each other, they couple instead to the electrostatic

field produced by the two nuclear charges. This situation is shown in Figure

IV.6(a) and is referred to as Hund’s case (a).

L

is so strongly coupled to the electrostatic field and the consequent frequency of

precession about the internuclear axis is so high that the magnitude of L

is not

defined: in other words L is not a good quantum number. Only the component ħ

of the orbital angular momentum along the internuclear axis is defined, where the

quantum number can take the values

Λ 0,1,2,3,…

(IV.26)

All electronic states with > 0 are doubly degenerate. Classically, this

degeneracy can be thought of as being due to the electrons orbiting clockwise

or anticlockwise around the internuclear axis, the energy being the same in

both cases. If = 0 there is no orbiting motion and no degeneracy.

The value of , like that of L in atoms, is indicated by the main part of the

symbol for an electronic state which is designated . . .

corresponding to 0, 1, 2, 3, 4, . . . . The letters . . are the Greek

equivalents of S, P, D, F , G, . . . used for atoms.

The coupling of S

to internuclear axis is caused not by the electrostatic field,

which has no effect on it, but by the magnetic field along the axis due to the

orbital motion of the electrons. Figure IV.6(a) shows that the component of S

along the internuclear axis is ħ. The quantum number S is analogous to MS in

an atom and can take the values

Σ , 1, … ,

(IV.27)

S remains a good quantum number and, for states with > 0, there are (2S+1)

components corresponding to the number of values that S can take.

11

The multiplicity of (2S+1) and is indicated, as in atoms, by a pre-superscript as,

for example, in 3 .

The component of the total (orbital plus electron spin) angular momentum

along the internuclear axis is ħ, shown in Figure 7.16(a), where the quantum

number is given by:

Ω |Λ Σ| (IV.28)

Since = 1 and =1,0,- 1 the three components of 3 have Ω= 2, 1, 0 and

are symbolized by 3 3 3

Spin–orbit interaction splits the components so that the energy level after

interaction is shifted by:

∆ ΛΣ (IV.29)

where A is the spin–orbit coupling constant. The splitting produces what is called

a normal multiplet if the component with the lowest has the lowest energy (i.e.

A positive) and an inverted multiplet if the component with the lowest has the

highest energy (i.e. A negative).

For states there is no orbital angular momentum and therefore no resulting

magnetic field to couple S to the internuclear axis. The result is that a state has

only one component, whatever the multiplicity.

Hund’s case (a), shown in Figure IV.6(a), is the most common case but, like all

assumed coupling of angular momenta, it is an approximation. However, when

there is at least one highly charged nucleus in the molecule, S-O coupling may be

sufficiently large that Land S

are not uncoupled by the electrostatic field of the

nuclei. Instead, as shown in Figure IV.6(b), Land S

couple to give J

, as in an

atom, and J couples to the internuclear axis along which the component is ħ .

This coupling approximation is known as Hund’s case (c), in which is no

longer a good quantum number. The main label for a state is now the value of .

This approximation is by no means as useful as Hund’s case (a) and, even when

it is used, states are often labeled with the value that would have if it were a

good quantum number.

For atoms, electronic states may be classified and selection rules specified entirely

by use of the quantum numbers L, S and J. In diatomic molecules the quantum

numbers and are not quite sufficient. We must also use one (for

heteronuclear) or two (for homonuclear) symmetry properties of the electronic

wave function ψe .

The first is the ‘g’ or ‘u’ symmetry property, which indicates that ψe is

symmetric or antisymmetric respectively to inversion through the centre of the

molecule. Since the molecule must have a centre of inversion for this property to

apply, states are labeled ‘g’ or ‘u’ for homonuclear diatomics only. The property

is indicated by a post-subscript .

The second symmetry property applies to all diatomics and concerns the

symmetry of ψe with respect to reflection across any (v) plane containing the

internuclear axis. If ψe symmetric to (i.e. unchanged by) this reflection the

state is labeled “ + ” , while if antisymmetric to this reflection it is labeled “-“.

This symbolism is normally used only for states.

IV.2.3 Electronic selection rules

Electronic transitions are mostly of the electric dipole type (electric

quadrupole or magnetic dipole also occur but are orders of magnitude

weaker in intensity). The selection rules for electric dipole transitions are:

• ∆Λ 0, 1 (IV.30)

For example, , , transitions are allowed but not or

• ∆ 0 (IV.31)

As in atoms, this selection rule breaks down as the nuclear charge increases. For

13

example, triplet–singlet transitions are strictly forbidden in H2 but not in

CO (although they are weak).

• ∆Σ 0;∆Ω 0, 1 (IV.32)

for transitions between multiplet components

•+⇹ ;+↔+; -↔ - (IV.33)

This is relevant only for transitions.

• g↔u; g⇹g; u⇹u (IV.34)

For Hund’s case (c) coupling (Figure IV.6b) the selection rules are slightly

different. Those in Equations (IV.31) and (IV.34) still apply but, because and

are not good quantum numbers that in Equation (IV.30) and Equation (IV.32) do

not. In the case of Equation (IV.33) the +/- signs refer to the symmetry of ψe to

reflection in a plane containing the internuclear axis only when

IV.2.4 . Vibrational coarse structure

IV.2.4.1 Potential energy curves in excited electronic states

Potential energy curves derive from the Born-Oppenheimer approximation and

represent the potential due to electrons in which nuclear motion occurs. Figure

IV.7 shows a typical curve with a potential energy minimum at re, the

equilibrium internuclear distance. Dissociation occurs at higher energy and the

dissociation energy is De , relative to the minimum in the curve, or D0, relative

to the zero-point level (v0= level). The vibrational term values, G(v), for an

anharmonic oscillator are given by

⋯ (IV.35)

Where e is the harmonic constant and e xe is the anharmonic constant. This expression

can be further expanded but the additional terms are very small and contribute little to the

total energy.

For each excited electronic state of a diatomic molecule there is a potential

energy curve and, for most states, the curve appears qualitatively similar to that

in Figure IV.7.

As an example of such excited state potential energy curves, Figure IV.8 shows

curves for several excited states and also for the ground state of the short-lived

C2 molecule. The ground electron configuration is

1 ∗1 2 ∗2 2 (IV.36)

giving the gX 1 ground state. The low-lying excited electronic states in Figure

IV.8 arise from configurations in which an electron is promoted from the u 2p

or u 2s orbital to the g 2p orbital. The information contained in Figure IV.8

has been obtained using various experimental techniques to observe absorption

or emission spectra. The combination of techniques for observing the spectra,

including a high-temperature furnace, flames, arcs, discharges and, for a short-

lived molecule, flash photolysis is typical.

In addition to these laboratory-based experiments it is interesting to note that

some of the bands of C2 are important in astrophysics. The so-called Swan bands

have been observed in the emission spectra of comets and also in the absorption

spectra of stellar atmospheres, including that of the sun, in which the interior of

the star acts as the continuum source.

When C2 dissociates it gives two carbon atoms which may be in their ground or

excited states. We saw, earlier that the ground configuration 1s22s22p2 of the C

atom gives rise to three terms: 3 P, the ground term, and 1 D and 1 S, which are

successively higher excited terms. Figure IV.8 shows that six states of C2, including

the ground state, dissociate to give two 3 P carbon atoms. Other states give

dissociation products involving one or both carbon atoms with 1 D or 1 S terms.

Just as in the ground electronic state a molecule may vibrate and rotate in

excited electronic states. The total term value T for a molecule is (IV.37)

15

with the electronic term value Te, corresponding to an electronic transition

between equilibrium configurations and, G(v) and F(J) are the vibrational and

rotational term values, resp. .

The vibrational term values for any electronic state, ground or excited, are given

by Equation (IV.35), where the constants vary from one electronic state to another.

Figure IV.8 shows that the equilibrium internuclear distance re is also different for

each electronic state.

IV.2.5.2 Progressions and sequences

Figure IV.9 shows sets of vibrational energy levels associated with two

electronic states between which we shall assume an electronic transition is

allowed. The vibrational levels of the upper and lower states are labeled by the

quantum number v’ and v’’; respectively. We discuss absorption as well as

emission processes and it will be assumed, unless otherwise stated, that the lower

state is the ground state.

In electronic spectra there is no restriction on the values that v can take but, as

we shall later, the Franck–Condon principle imposes limitations on the intensities

of the transitions. Vibrational transitions accompanying an electronic transition

are referred to as vibronic transitions. These vibronic transitions, with their

accompanying rotational (i.e., rovibronic transitions), give rise to bands in the

spectrum, and the set of bands associated with a single electronic transition is

called an electronic band system. Vibronic transitions may be divided

conveniently into progressions and sequences. A progression, as Figure IV.9

shows, involves a series of vibronic transitions with a common lower or upper

level.

Quite apart from the necessity for Franck–Condon intensities (see below) of

vibronic transitions to be appreciable, it is essential for the initial state of a

transition to be sufficiently highly populated for a transition to be observed.

Under equilibrium conditions the population Nv’’ of any v’’ level is related to that

of v’’=0 via:

0 (IV.38)

which follows from the Boltzmann equation.

Because of the relatively high population of the v”= 0, the v”=0 progression is

likely to dominate the absorption spectrum.

In emission the relative populations of the v’ levels depend on the method of

excitation. In a low-pressure discharge, in which there are not many collisions to

provide a channel for vibrational deactivation, the populations may be somewhat

random. However, higher pressure may result in most of the molecules being in

the v’=0 level and its associated progression is prominent. Nowadays, the method of

choice for excitation is monochromatic (often laser) light, which can selectively

populated specific vibronic levels.

The progression with v”=1 may also be observed in absorption but only in

molecules with a vibrational frequency low enough for the v”=1 level to be

populated at kT. This is the case for example with iodine for which 0=213 cm-1

(while kT= 203 cm-1 at T=293 K). As a result the gXBu

1

0

3 visible transition

shows in absorption at RT, not only the v”=0 progressions but also the v=1 and 2

ones (see figure IV.10).

Fluorescence= transitions that are spin-allowed. Their lifetimes are in the

nanosecond time domain.

Phosphorescence= spin-forbidden transitions, that however occur due to spin-

orbit coupling which relaxes the spin selection rule. These transitions have long

lifetimes in the 100s of ns to the msec time domain.

Consequently, there is a tendency to distinguish the two processes according to

their lifetimes. One has to be careful, in cases of strong spin–orbit coupling, as the

lifetime of a transition between states of differing multiplicity may be relatively

short, leading to an erroneous description as fluorescence.

17

If emission is from only one vibrational level of the upper electronic state it is

referred to as single vibronic level fluorescence (or phosphorescence).

A group of transitions with the same value of v is referred to as a sequence.

Because of the population requirements long sequences are observed mostly in

emission.

Progressions and sequences are not mutually exclusive (see fig. IV.9). Each

member of a sequence is also a member of two progressions. However, the

distinction is useful because of the nature of typical patterns of bands found

in a band system.

Progression members are generally widely spaced with approximate separations

of e ’ in absorption and e ” in emission. In contrast, sequence members are more

closely spaced with approximate separations of e ’ - e ”.

The general symbolism for indicating a vibronic transition between an upper and

lower level with vibrational quantum numbers v’ and v”, respectively, is v’-v”,

consistent with the general spectroscopic convention. Thus the pure electronic

transition (i.e. not involving vibrations) is labeled 0–0.

IV.2.5.3TheFranck–Condon principle

In 1925, before the development of the Schrödinger equation, Franck put forward

classical arguments to explain the various types of intensity distributions found

in vibronic transitions. These were based on t h e B o r n - O p p e n h e i m e r

a p p r o x i m a t i o n , i . e . , electronic motion is much more rapid than nuclear

motion, so during an electronic transition, the nuclei have very nearly the same

position and velocity before and after the transition.

Possible consequences of this are illustrated in Figure IV.11a, which shows

potential curves for the lower state, which is the ground state if we are

considering an absorption process, and the upper state. The curves have been

drawn so that re’ > re”. In absorption, from point A of the ground state in Figure

IV.11(a) (zero-point energy can be neglected in considering Franck’s semi-

classical arguments) the transition will be to point B of the upper state. The

requirement that the nuclei have the same position before and after the transition

means that the transition is between points which lie on a vertical line in the

figure: this means that r remains constant and such a transition is often

referred to as a vertical transition. The second requirement, that the nuclei have

the same velocity before and after the transition, means that a transition from A,

where the nuclei are stationary, must go to B, as this is the classical turning point

of a vibration, where the nuclei are also stationary. A transition from A to C is

highly improbable because, although the nuclei are stationary at A and C, there is

a large change of r. An A to D transition is also unlikely because, although r is

unchanged, the nuclei are in motion at the point D.

Figure IV.11b illustrates the case where re’ ≈ re”. Here the most probable transition

is from A to B with no vibrational energy in the upper state. The transition

from A to C maintains the value of r but the nuclear velocities are increased due

to their having kinetic energy equivalent to the distance BC.

In 1928, Condon treated the intensities of vibronic transitions quantum

mechanically. The intensity of a vibronic transition is proportional to the square of

the transition moment evR

, which is given by Equation II.13:

evevevev dR "'* (IV.39)

where

is the electric dipole moment operator and the ψ’s are the vibronic

wave functions of the upper and lower states, respectively. The integration is over

electronic and vibrational coordinates. Within the Born–Oppenheimer

approximation, they can be factorized in an electronic and a vibrational part, and

eq. IV.39 becomes:

drRddR vveveveveev'''*'''''*'*

(IV.40)

With

19

eeee dR "'* (IV.41)

is the electronic transition moment.

As a consequence of the BO approximation, the electronic part is independent of the

nuclear coordinates and we take it outside the integral in Equation (IV.40), regarding

it as a constant. The quantity drvv'''* is called the vibrational overlap function.

Its square is known as the Franck–Condon factor to which the intensity of the

vibronic transition is proportional. In carrying out the integration the requirement

that r remain constant during the transition is necessarily taken into account.

The classical turning point of a vibration, where nuclear velocities are zero, is

replaced in quantum mechanics by a maximum, or minimum, in ψv near to this

turning point.

Figure IV.11 illustrates a particular case where the maximum of the v’=4 wave

function near the classical turning point is vertically above that of the v”=0 wave

function. Maximum overlap is indicated by the vertical line but appreciable

contributions extend to values of r within the dashed lines. Clearly, overlap

integrals for v’ close to four are also appreciable and give an intensity

distribution in the v”=0 progression with a maximum at v’=4.

The Franck-Condon intensity distribution for a progression starting from v=0

(either in absorption or emission) is strongly dependent on the relative

equilibrium distances of the two potential curves. This is illustrated in figure

IV.13. In the left panel, r’e≈ r”e and the progression starts with the 0-0

transition as the most intense, since the v”=0 and v’=0 wave functions have

maximum overlap. 4–0 transition is the most probable. In the middle panel r’e<r”e

and the maximum overlap is between v”=0 and v’=2, so the progression has a

maximum intensity at the 0-2 transition, although the other levels show

appreciable overlap too. Finally, for the right panel, r’e<<r”e and the maximum

overlap is close to the dissociation limit of the upper curve and the progression of

discrete bands terminates and the absorption continues into the dissociation

continuum giving rise to a continuous band.

IV.2.5.3 Dissociation energies

Experimentally the dissociation energy D0 can only be measured relative to the

v=0 level (Figure IV.14):

2/10 vv

GD

(IV.42)

If a sufficient number of vibrational term values are known in any electronic

state, D0 can be obtained by plotting Gv+1/2 as a function of v+1/2 (Figure

IV.15), where

)()1(2/1 vGvGGv (IV.43)

If all anharmonic constants except exe are neglected in equ. IV.42, Gv+1/2 is a

linear function of v and D0 is the area under the dashed curve in figure IV.15. in

most cases, only a few G values are known and a linear extrapolation to

G=0 has to be made, which called the Birge–Sponer extrapolation. This

only gives an approximate value of the dissociation energy, and since most plots

deviate from linearity, this delivers an overestimate of D0 (Fig. IV.14). Whether

this can at all be done depends very much on the relative dispositions of the

various potential curves in a particular molecule and whether electronic

transitions between them are allowed. How many ground state vibrational term

values can be obtained from an emission spectrum is determined by the FC

principle. To obtain an accurate value of D0 for the ground electronic state is

virtually impossible by vibrational (i.e. IR) spectroscopy because of the problems

of a rapidly decreasing population with increasing v. In fact, most

determinations are made from electronic emission spectra from one, or more,

excited electronic states to the ground state.

21

Obtaining Do’ for an excited electronic state depends on observing progressions

either in absorption, usually from the ground state, or in emission from higher

excited states. Again, the length of a progression limits the accuracy of the

dissociation energy.

If the values of re in the combining states are very different the dissociation

limit of a progression may be observed directly as an onset of diffuseness.

However, the onset is not always very sharp (see e.g., the spectrum of I2 in Figure

IV.10).

Figure IV.14 shows that

atomicit DD ~"~~00

'0lim (IV.35)

Hence, D’0 can be obtained from itlim~ if 0

~ the wavenumber of the 0–0 band, is

known.

Figure IV.10 shows that extrapolation may be required to obtain 0

~ , limiting the

accuracy of D’0.

Equation (IV.35) also shows that D”0 may be obtained from itlim~ since atomic~

is the wave number difference between two atomic states, the ground state 2

P3/2 and the first excited state 2P1/2 of the iodine atom, known accurately from the

atomic spectrum. Thus the accuracy of D”0 is limited only by that of itlim~ .

D’e and D”e, the dissociation energies relative to the minima in the potential

curves, are obtained from obtained D0’ and D0”

)0(GDDe (IV.36)

where G(0)is the zero-point term value.

IV.2.5.6 Repulsive states and continuous spectra

The ground configuration of the He2 molecule is, according to Figure IV.16,

22 )1*()1( ss ug and is expected to be unstable because of the cancelling of the

bonding character of a )1( sg orbital by the antibonding character of a )1*( su

orbital. The potential energy curve for the resulting X 1+ state shows no

minimum but the potential energy decreases smoothly as r increases, as shown in

Figure IV.16(a). Such a state is known as a repulsive state since the atoms repel

each other. In this type of state, either there are no discrete vibrational levels or

there may be a few in a very shallow minimum. All, or most, of the vibrational

states form a continuum of levels.

Promotion of an electron in He2 from the )1*( su to a bonding orbital produces

some bound states of the molecule of which several have been characterized in

emission spectroscopy (e.g. the A-X transition in fig. IV.16a or the a-b transition in fig. IV.16b).

23

Figure IV.1: Overlap of s and p atomic orbitals.

Figure IV.2 Formation of two molecular orbitals from two identical AOs. EA is the

binding energy of the AO.

Figure IV.3 Formation of molecular orbitals from 1s, 2s, and 2p atomic orbitals

25

Figure IV.4: Molecular orbital energy level diagram for first-row homonuclear

diatomic molecules. The 2px, 2py, 2pz atomic orbitals are degenerate in an atom and

have been separated for convenience. (In O2 and F2, the order of g 2p and u 2p is

reversed.)

Figure IV.5: In HCl (a) the single-bond molecular orbital is formed by a linear

combination of 1s on H and 3pz on Cl, and (b) electrons in the 3px and 3py atomic

orbitals on Cl remain as lone pairs.

Figure IV.6: (a) Hund’s case (a) and (b) Hund’s case (c) coupling of orbital and

electron spin angular momenta in a diatomic molecule

Figure IV.7: Potential energy curves. Dashed line: harmonic potential, full line:

anharmonic potential, where De is the energy of dissociation measured from the

minimum of the curve and D0 is the energy of dissociation from the v=0 level,

27

Figure IV.8: Potential energy curves for the ground and several excited states of C2. The

ground state is indicated by X, while the excited state of similar multiplicity are indicated

by capital letters, those of a multiplicity different to that of the gs are indicated by small

letters.

Figure IV.9: Vibrational progressions and sequences in the electronic spectrum of a

diatomic molecule.

Figure IV.10: Progressions with v”=0, 1 and 2 in the gXB

u

1

0

3

system of gaseous I2

at room temperature. The v”= 1 and 2 levels are populated as a result of the Boltzmann

equilibrium in the room temperature vapour.

29

Figure IV.11: Illustration of the Franck principle for (a) r’e> r”e ; (b) r’e > r”e. The

vibronic AB transition is the most probable in both cases.

Figure IV.12: Franck–Condon principle applied to a case in which r’e> r”e and the 4–0

transition is the most probable.

31

Figure IV.13: general shape of Franck–Condon progression starting from v”=0 (i.e.

absorption) for different configurations of the ground and excited states.

Figure IV.14: Dissociation energies D’0 and D”0 may be obtained from limit, the onset of a

continuum in a progression of a diatomic molecule (I2 in the present case).

Figure IV.15: Birge-Sponer extrapolation (dashed line) for determining the dissociation

energy relative to the v=0 level. Typically, the measured curve is the solid one.

33

Figure IV.16: (a) The repulsive ground state and a bound excited state of He2. (b) Two bound

states and one repulsive state of H2.